Paper · Oct 24, 2025

Ricci Flow on ALF manifolds

Dain Kim, Tristan Ozuch

math.DGarXiv:2510.21997
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Abstract

We prove that on ALF -manifolds with the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's -functional, we define a renormalized functional whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF -metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of to positive relative mass statements for ALF metrics.

Comments: 40 pages

Ricci flowRicci-flat & Calabi–YauKähler geometryStability & uniqueness

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    Isaac M. Lopez and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Tristan Ozuch · Shared terms “renormalized”, “hyperkähler”, “Perelman”

  2. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Tristan Ozuch · Shared terms “along Ricci”, “linear stability”, “unstable”

  3. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness

  4. Linear stability and instability of Kähler Ricci solitons

    Keaton Naff and Tristan Ozuch PDF

    Topics Kähler geometry, Stability & uniqueness · Same author Tristan Ozuch · Shared terms “linear stability”, “unstable”

  5. The asymptotic behavior of the steady gradient Kähler-Ricci soliton of the Taub-NUT type of Apostolov and Cifarelli

    Daheng Min PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “ALF”

  6. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang PDF

    Topics Ricci-flat & Calabi–Yau · Shared terms “along Ricci”

  7. HKT geometry on locally conformally hyperkähler manifolds

    Giovanni Gentili and Luigi Vezzoni PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “hyperkähler”, “conformally”

  8. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “unstable”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.